Central limit theorem for eigenvectors of heavy tailed matrices

Abstract

We consider the eigenvectors of symmetric matrices with independent heavy tailed entries, such as matrices with entries in the domain of attraction of α-stable laws, or adjacency matrices of Erdös-Rényi graphs. We denote by U = [uij ] the eigenvectors matrix (corresponding to increasing eigenvalues) and prove that the bivariate process converges in law to a non trivial Gaussian process. An interesting part of this result is the rescaling, proving that from this point of view, the eigenvectors matrix U behaves more like a permutation matrix (as it was proved in [17] that for U a permutation matrix, is the right scaling) than like a Haar-distributed orthogonal or unitary matrix (as it was proved in [18, 5] that for U such a matrix, the right scaling is 1).